Less is more: Granularity of information, estimation errors and optimal portfolios

By introducing the world to mean-variance optimization (MVO), Markowitz (1952) laid out the fundamental ideas that created and shaped modern finance theory. The main goal of modern portfolio theory (MPT) is simple: Once an investor has formed believes about future return and risk of different assets, he should use these believes together with the MVO-approach to generate an asset allocation that maximizes the expected return of a portfolio given a predefined level of risk. Today, nearly seven decades after its first publication, MVO remains one of the most significant models for explaining how rational investors should behave, for how optimal portfolios can be constructed and for formalizing the benefits of diversification (Michaud, 1989). Furthermore, the resulting notion of the efficient frontier heavily influenced the research of Sharpe (1964), Lintner (1965) and Mossin (1966) and thereby significantly contributed to the development of the Capital Asset Pricing Model (CAPM) and subsequent asset pricing models.

Despite the fact that MVO has a sound theoretical background, a compelling, straight-forward narrative and has been around since the early 1950’s, investment practitioners have been reluctant when it comes to implementing the approach in a real-world portfolio setting. The reasons for why investment professionals shy away from practically adapting MVO are manifold. Up front, the structural organization of institutional investors alongside their investment processes and committees are oftentimes not compatible with purely quantitative investment approaches such as MVO (Michaud, 1989). In addition to that, the majority of asset managers have traditionally managed their client’s portfolios on a discrete and informal basis leading to the notion that only “skilled” portfolio managers could be successful at the “craft” of portfolio management. Furthermore, and in contrast to its intuitive theory, the output of the optimization approach (i.e. the optimized asset weights) are often not intuitive and seemingly lack practical investment value as MVO portfolios have a tendency for poor out-of-sample performance (Michaud, 1989). Often, unconstrained MVO portfolios are outperformed by the naively diversified portfolio (the equally weighted or 1/N-portfolio, see e.g. Michaud, 1989).

Its unattractive out-of-sample track-record is amongst the most significant reasons for the timid adoption of MVO in the investment industry and is mostly driven by its tendency to overweight estimation errors. More precisely, MVO aims at achieving an optimal trade-off between expected portfolio return and risk, which according to Michaud (1989) implies, that assets with high (low) expected returns, low (high) variance and negative (positive) covariance with other assets are overweighted (underweighted). As a higher return typically comes with higher risk, these assets are usually also the ones with the most severe estimation errors (Michaud, 1989). Consequently, MVO does not optimize return per unit of risk but rather estimation errors. In addition to that, MVO often yields weights that are counter-intuitive and extreme, resulting in very concentrated portfolios which causes further performance issues (Frost and Savarino, 1988; Michaud, 1989). Furthermore, the fact that MVO does not support investors at coming up with precise and accurate estimates for future performance and risk but treats the estimates as if they were a 100% correct and precise further adds to its estimation error maximizing nature. Moreover, Best and Grauer (1991) argue that the returns, variance and weights of a portfolio are highly sensitive towards relatively small changes in an asset`s expected return. Kolm et al. (2014) later confirm this observation and add that the uncertainty linked to the estimation error has a stronger influence on the returns, variance and weights of an MVO portfolio compared to flawed estimates of the covariance matrix.

Previous attempts to reduce the impact of estimation errors have had limited success. In one of the most elaborate efforts to date, for example, DeMiguel et al. (2009) compare various techniques that aim at reducing the harmful impact of estimation errors only to conclude that many optimization models that have deliberately been designed to deal with flawed parameter inputs do not consistently improve portfolio performance due to the heavily distorting effect estimation errors have on optimized portfolio weights. The authors further state that estimating reliable parameter inputs solely on the basis of historical data is notoriously difficult as it requires vast amounts of data (DeMiguel et al., 2009). For most assets the existing data history is not long enough and thus investors are forced to use less observations to estimate the necessary parameter inputs. As a consequence, the estimated parameters are not reliable and come with large estimation errors. Ultimately, the heavy impact of estimation errors on the out-of-sample performance of MVO portfolios raises the question whether or not asset-specific information based on historical data is worth incorporating when optimizing a portfolio as empirical evidence suggests that no asset-specific information yields higher economic value to investors. In contrast to that, Allen et al. (2019) demonstrate, that MVO portfolios which use expected returns and (co-)variances that have been forecasted with established predictor variables as parameter input can significantly outperform naively diversified portfolios in terms of net return and Sharpe ratios.

All in all, two contradicting lessons can be learned: On the one hand, the out-of-sample performance of mean-variance portfolios that have been optimized on the basis of historical data suggests abandoning MVO in favor of a naively diversified weighting scheme (i.e. DeMiguel et al., 2009). On the other hand, according to Allen et al. (2019) investors can use forecasted returns as approximations for expected returns and thereby improve the out-of-sample performance of MV-optimized portfolios, given a modest predictive power (i.e. an out-of-sample R^2 of 0.5%).

Naturally, these opposing results lead to the question of whether or not there is an ideal level of informational content at which the performance benefits that are due to incorporating predicted returns in MVO are large relative to the harming impact of flawed estimates. Surprisingly, academic research revolving around this question is nonexistent which in turn implies a large prevailing research gap. Motivated by this lack of research, this paper examines the interplay between various degrees of informational content of forecasted inputs and how this affects the estimation error and out-of-sample performance of MVO portfolios. This paper therefore investigates how the sequential reduction of informational content of forecasted equity returns influences the performance of MVO portfolios. We show that step by step reducing the informational content of prediction-based inputs for expected returns via an application of relative ranks reduces the impact of estimation errors further and further while still maintaining levels of information that outperform the 1/N-portfolio.

Reasons for this are manifold: Early research of Frost and Savarino (1988) as well as Michaud (1989) has pointed out that extremely promising expected return and risk characteristics of assets are typically unreliable as they are mostly caused by estimation errors. If investors still choose to rely on these flawed parameter inputs, extreme weights and concentrated portfolios are often the result alongside a poor out-of-sample performance. By reducing the informational content of estimated or forecasted optimization inputs from exact forecasts to ranks, some of the extreme differences within the cross-section of assets caused by estimation errors are also dropped, which reduces estimation error and in turn should result in less extreme weights and less concentrated portfolios. Generally speaking, our research contributes to the intersection between the research related to handling estimation errors, Bayesian-extensions that allow the incorporation of alternative information such as ordinal or qualitative data and return predictability. Our idea of reducing the informational content of input parameters further contributes to the literature in that it offers a novel way of reducing the impact of estimation errors while still making use of forecasted means as optimization inputs.

We find that our approach (significantly) outperforms the plug-in mean-variance portfolios as well as the naively diversified portfolios on a risk-adjusted basis which highlights the benefits of reducing informational content of input parameters. Furthermore, we present evidence that suggests that our approach is more effective when estimation errors are expected to be larger, i.e. when the cross-section is larger. The robustness of these results is supported by the fact that our approach yields almost identical results when bootstrapped with randomly drawn samples of increasing sizes.